Week 1? Flashcards
(397 cards)
1
Q
Extensive properties
A
Depends on amount
2
Q
Intensive property
A
Independent of amount
3
Q
Determine whether extensive
A
- Additive e.g if property is V^2, (V^a)^2 + (V^b)^2 should = (V^a + V^b)^2
- Proportional to size of quantity of substance
4
Q
Determine whether intensive
A
- Does not add up when part of system are combined
- Constant regardless of substance size
5
Q
Define thermodynamic state
A
Set of conditions (T, P,V) that’s fully describe system at a given moment
6
Q
Equilibrium
A
All macroscopic properties of a system are well-defined and remain constant over time
7
Q
Response function
A
how a system reacts to changes in external variables
8
Q
Temperature
A
Measure of average kinetic energy of particles in a system
9
Q
function of state
A
Any thermodynamic property that depends only on the state of the system and not the history of how it reached that system
10
Q
Correct Reciprocity theorem in words
A
Swapping positions of variables agives same result- symmetry
Force at A causing displacement at B, will cause same displacement at A when the force is applied at B
11
Q
Correct: Reciprocal theorem in words
A
Response of one system to external force is antisymmetric - e.g. PV
12
Q
Fundamental relation
A
Equation that relates internal energy to state variable
13
Q
System
A
Portion of universe being studied
14
Q
Reversible
A
- Infinitely slow and at thermodynamic equilibrium at every stage
- Can be exactly returned to original state by reversing process
15
Q
Features of reversible
A
- No entropy generation
- Infinitely slow
- Idealised
16
Q
Quasi static
A
- So slow that can be assumed to be in equilibrium at all times
17
Q
Features of quasi static
A
- Slow and continuous
- No sudden changes
- Approximates reversible
18
Q
Irreversible
A
Cannot be reversed by an infinitesimally small change in external conditions
19
Q
Features of irreversible
A
- Entropy generation
- Not reversible
- Energy dissipated
20
Q
bulk modulus
A
Property that measures a material’s resistance to uniform compression
21
Q
Formula bulk modululs
A
K = -V * ΔP/Δ V
22
Q
Improper differentials:
A
- Discontinuous e.g. phase transition
- Sharp corners/edges e.g. critical temp/pressure
- Singularities e.g. compressibility near critical point
- Path dependent
23
Q
Specific properties
A
Extensive property/mass
24
Q
Relate density and volume
A
Density = 1/volume
25
Cycle
System that returns to original state
26
Celsius to Fahrenheit
X1.8 +32
27
Fahrenheit to Rankine
+459.67
28
Rankin to Kelvin
X1.8
29
Kelvin to Celsius
+273.15
30
Absolute temperature
K, R = start at zero
31
Absolute pressure
Actual pressure + gauge pressure
32
Atmospheric pressure
1 atm = 101.325 kPA
33
How to find conversion from Rankin to Kelvin
- both absolute zero (no y intercept needed)
- m = e.e.g freezing point in F/freezing point in C
=491.67/273.15 = 9/5
34
When saying “twice as hot”
Need to reference temperature scale or does not make sense
35
Relaxation time
Time needed to come to thermal equilibrium
~2minutes
36
Thermal expansion coefficient, β
(ΔV / V) / ΔT
37
What is room temperature?
25 Celsius
38
Boltzmann constant in J/K
1.381x10^-23
39
Boltzmann constant in L.atm/(mol.K)
0.0821
40
How to approach problems asking to compare e.g. number of moles in 2 rooms
Two simultaneous equations based on ideal gas law
- Find something to equate
- Rearrange to equate as e.g.Na = X Nb
41
What is a good estimate of average particle speed?
Vrms
42
Vrms formula
SQRT (3Kt)/m
NB Mass = Atomic mass/avogadro’s in GRAMS and Kb in KG -> need to convert and check SI units/dimension analysis
43
Which molecules move faster in gas in thermal equilibrium
= 1/2 m v^2 , so one with lowest mass
44
Pressure due to KE of particles
= Force over area, where force = rate of change in momentum
= Nmv^2 /V
45
Effusion
Gas molecules pass through tiny hole without collisions between molecules
46
Graham’s law of effusion
Rate of effusion ∝ 1/sqrt (M)
47
Change in velocity in x direction after particle collision
= -2 * velocity in x direction
(ΔV = Vf - Vi = -Vi - Vi = -2 Vi )
48
number of particles colliding with surface of area A
Pressure =force/area where force =rate of change in momentum and v = -2v due to collision
P = m2Nvbar/ (A Δt)
->rearrange for N
49
What is the miracle of thermodynamics?
Universality of thermodynamics
50
Extensive properties
V,n
51
Intensive properties
P, T
52
Examples of reversible reactions
Isothermal expansion/compression
Phase change at equilibrium
53
Examples of irreversible
- Spontaneous
- Free expansion (e.g Joule expansion)
-
54
Features of free expansion
1. No external work
2. No heat exchange (insulated)
3. U constant
4. Irreversible
5. Entropy rise
55
Which are state functions?
U, H, S, G, A/F,
56
zeroth Law
If 2 systems are in THERMAL equilibrium with a third, then they are in equilibrium with each other
57
Which are important non state functions?
Heat, work
58
Negative sign for work
Work done on system i.e. compression
59
Positive sign for work
Work done by system i.e. expansion
60
Miracle of thermodynamics
Ability to describe thermodynamic behaviours of systems ignoring that they are made of molecules and focus on variables only
61
Boyle’s law
PV =constant
62
Charles’s Law
V/t = constant
63
Gay-Lussac Law
P/t = constant
64
Avogadros Law
V/n= constant
65
How to find isothermal work from isobaric?
1. Work isobaric = P1 (V2-V1)
2. Work isothermal. = nRT1 ln (V2/V1)
3. Sub P1 = nRT1/V1 into 1
- Work isobaric = nRT/V1 ( V2 - V1) = nRT1(V2/V1 1)
4. Rearrange 3:
- ΔV = (Work isobaric/nRT1) +1
5. Sub 4 into 2
- Work isothermal = nRT1 ln (Work isobaric/nRT1) +1
66
Calculate work from graph
Area under curve (down to x axis)
67
Microstate
Microscopic information about a state e.eg position and momenta of molecules
68
Macro state
Described by its macroscopic properties
- Many microstate configurations
69
What makes it possible to measure temperatures?
Zeroth law
70
Define thermometer
Working substance with measureable property - length, pressure etc, which changes in a regular way as substance becomes hotter/colder
71
Pressure at triple state of water
610.6Pa
72
Water freezing and boiling in Fahrenheit
32 and 212
73
function of state?
Only depend on its current thermodynamic state, not path
74
PV units for ideal gas law 8.314J/mol.K?
P = Pa
V = m^3 (i.e. litres/1000)
75
PV units for ideal gas law 0.0821 L.atm/mol.K?
P = atm
V = L
76
Find function of exact differential
1. Verify exact by dx/dy = dy/dx
2. Integrate e.g. dx/dy wrt y (or dy/dx wrt x)
3. Compare integrand to e.g. dx in differential
4. Integrate what is left wrt e.g. x
5. State function
77
More insulating in series or parallel? Why?
- Better insulation in series as added as resistances
( Series = R1+ R2+…; 1/ Parallel = 1/R1 + 1/R2+….)
78
How to calculate heat current
Q/t = (kA ΔT)/d
D= thickness
79
Reduction in heat loss due to huddling in penguins
1. Model penguins as cylinders
2. Subsitiute r in terms of area (a = π r^2)
3. Relate the 2 (heat loss penguin / heat loss cylinder*N)
4. Should have 2 similar equations with extra factor of SQRT (1/N) on top ->factor out to have 1/4th root of N
80
Calculate gas pressure inside cylinder with piston
Pgas = Patm + mg/A
81
Convert bar to PA
1 bar = 100,000 Pa
82
Work for gas in cylinder with piston
W = F.d = (mg + Patm.A) * ( ΔV/A)
= ( mg/A + Patm) * ΔV
83
Way to compute work for adiabatic
Work = (P1V1 - P2V2)/(γ-1)
84
Where is heat in / out on PV
QH (in) top horizontal and left
QC (out) bottom horizontal and right
85
Which curve is higher I.E. more work on PV diagram - isothermal or adiabatic?
Isothermal
86
Is heat lost or gained during compression?
Lost
87
Efficiency
Work/Qin
Or Qin - Qout/ Qin
88
Define Newtons Law of Cooling
The hotter something is compared to its surroundings, the faster it cools down.
- the rate at which an object cools down is proportional to the difference between its temperature and the surrounding temperature.
89
Differential form of 1st law
- dU = TdS - PdV
90
First law for cyclical
W = U
91
Define heat capacity
amount of heat needed to raise the temperature of a substance by 1 degree Celsius (or 1 Kelvin).
92
Heat capacity formula
C = Q/ ΔT
93
Which is Qlow?
One for compression/pressure reduction
- decreased P and/or V
94
Which is Q high
One for expansion/higher pressure
95
Place of highest temperature on PV diagram?
Top left
96
Meaning of “1 unit mass” in question?
- 1kg of air contains n= 1/M moles
- M = molar mass e.g. 28.97g/mol for air
97
What is important to note in cycle heat calculations?
Isobaric or isochoric as changes heat equation used
98
If 35kPA is needed to move a piston on stops in a cylinder, what is P2?
35kPA
99
Formula to calculate gas pressure in piston cylinder set up
P gas= Patm + mg/A
100
Directions for force body diagram piston-cylinder
Only force from gas pressure points up, all others downswards
101
Formula for spring force
= ks x = ks (V - V 1) / A
102
Which heat is absorbed in engines?
QH
103
Which heat is rejected/emitted in engine?
QL
104
Conversion from hp
Hi x 0.7355kW/hp
105
Rate of fuel consumption
= QH/fuel burnt
106
Location of highest temperature from PV diagram
- From ideal gas law T = PV/nr
- Location of highest P and highest V GENERALLY
107
Work for adiabatic process
W = (P2V2 - P1V1)/ (1 - γ )
108
Adiabatic equations
PV γ
TV (γ-1)
P (1-Y) T γ
109
Entropy exchanged for adiabatic
Zero as ΔQ = 0
110
Formula for power of engine
P = QHigh x η / Δtime
111
One method to solve for a variable in adiabatic
- Take log
- Separate variables
- Bring down power
- Rearrange
112
How to integrate along straight one when proving path independence?
E.g. y=2x, dy = 2dx -> sub into dx part including limits
113
Creation of reversible isothermal process for proofs
Q = W = nrT ln (V2/V1)
114
Entropy change for isothermal
ΔS = Q/T
115
Entropy formula
ΔS = S2 - S1 = dQ/dT
116
Entropy change in engine for QH
ΔSh = QH/Th
117
Entropy change in engines for QL
ΔS L = QL/TL
118
ΔEntropy direction if heat transferred to system
Increases
119
Δ Entropy direction if heat lost from system
Decreased
120
Entropy in irreversible
Positive
121
Entropy in reversible
0
122
ΔEntropy between 2 reservoirs
ΔS = ΔSH plus ΔSL
123
Calculate COP
Inverse of efficiency
QH/W
124
Derive isothermal entropy
1. State entropy equation
2. Differential form of 1st law : dU = dQ - P dV
3. Isothermal -> ΔU = 0
4. Sub dQ = - PdV into 1
5 .Sub P = nRT/V into 5
6. Cancel and integrate
125
Derive entropy for free expansion i.e irreversible
1. Start with reversible, isothermal
2. ΔU = 0 so dQrev = PdV
3. Sub in P = nRT / V
4. Entropy is a state function so same for irreversible as reversible
5. Integrate nRT/TV dV
- Vi - x, Vf = (1-x)V so Vtotal = V
6. Should get nR ln(1/x)
126
Statistical form of entropy equation
ΔS = N Kb ln 2
127
Stochastic thermodynamics
Systems that are small and exhibit random, and therefore sometimes unexpected behaviour - part of statistical entropy
128
Most stable system in terms of entropy
- Maximum entropy
- Largest microstates
129
Boltzmann formula of entropy
S = Kb ln Ω
130
Entropy for gases mixing wiht volumes xV and (1-x) V
ΔS mix = Boltzmanns = N Kb ln 2
131
Keys features of free expansion
- No work done - expands into a vacuum so no external pressure to work against
- No heat exchange - expands into vacuum -> no heat source or sink in contact with system
132
Difference between irreversible and reversible entropy changes of universe
Irreversible is in a vacuum with no external pressure
- ΔS total = ΔS system + ΔS surroundings
- Irreversible -> Pext =0 -> dW = 0, dQ = 0
ΔS total = nR ln (V2/V1) + 0
- Reversible -> ΔS = 0
ΔS total = nR ln (V2/V1) - nR ln (V2/V1)
133
COP for refrigerator with temperature/heat only
Tc/ (Th-Tc)
134
Relate Carnot efficiency to COP heat pump
- n = 1 - heating
135
Carnot efficiency/COP: engine vs heat pump vs fridge
- Engine: 1 - Tc/Th
- Heat pump: 1 - Th/Tc OR 1 - Tc/Th
- Refrigerator: [Tc/ (Th-Tc) ] = COP = Qc/Work
136
Heat pump vs refrigerator
- Heat pump: heat AND cool
- Fridge: Only cools
137
Formula for work in engines
QH - QC
138
Proof that sum of Carnot efficiency for numerous reservoirs same as just for firsta and last
1. Create equations for:
A) Q2 = Q1(T2/T1)
B) n = 1 - Tc/Th = w/Qh
C) Work = nQh
2. Find sum of work
3. Sub all in and factor out Q1
4. One common denominator - > should mostly cancel
139
Velocity RMS
- location and formula
- Highest ( on right of Boltzmann graph)
- Sqrt ((3RT)/M)
NB M = molar mass, kg/mol
140
Reciprocal theorem
Reciprocity -> inverse
(Dx/dy)dz = 1/ (Dx/dy)dz
141
Reciprocity theorem
Cities - many
(Dx/dy)z (dy/dz)x (dz/dx)y = -1
142
Reciprocal theorem proof
1. Derivative of z as a function of x and y
2. Dz = (dz/dx)y = (dz/dy)x
3. Assume curve where z = constant so 2. = 0
4. Find (dx/dy)Z and (dy/dx)Z to cancel/equate
ALL HAVE ‘Z’ IN NUMERATOR
(dy/dx)z = - (dz/dx)y / (dz/dy)x AND
(dx/dy)z = - (dz/dy)x / (dz/dx)y
143
Reciprocity theorem derivation
1. Derivative of z as a function of x and y
2. Dz = (dz/dx)y = (dz/dy)x
3. Assume curve where z = constant so 2. = 0
4. Means (dx/dy)z = dx/dy -> sub into below
(dy/dx)z = - (dz/dx)y / (dz/dy)x AND
5. (Dz/dx)y (dx/dy)z dy + (dz/dy)x dy = 0
144
Isothermal entropy formula
ΔS = nR ln (Vf/Vi) = - nR ln (Pf/Pi)
(First from dQ = p dV = nRT/V dV)
145
Isochoric entropy formula
ΔS = n Cv ln (Tf/Ti)
146
Isobaric entropy formula
ΔS = nCP ln (Tf/Ti)
147
Remember about temperature when calculating total entropy change with reservoir
For reservoir:
- T res is a constant so comes out of integral
- Entropy change for reservoir is negative
-> Δ S = - n Cv (Tres- Ti)
148
Net entropy for Carnot
Zero -> reversible
149
Relevance of entropy for Carnot cycle =0
Temperature at end of adiabatic transformation = Temperature of reservoirs to perform isothermal transformation
150
How to calculate entropy in liquids of different mass mixing
- Need to equate 2 heats [Cp(Tf-Ti)] in terms of ratios of mass (CAREFUL WITH SIGNS) and solve for T3 -> the final heat
- Final entropy integral is of Cp dT/T -> ln
151
Relation of masses of 2 gases to Cv or Cp
A times mass = A x Cv = A x Cp
152
How to calculate entropy from PV diagram
1. First find work - can use AUC
2. Then find Q from 1st law as need to know whether this is negative so integrate in correct direction
OR if isochoric/isobaric can use heat capacity
153
Infinitesimal first law for 1 mole
DQ = Cv dT + p dV
154
How to draw TS diagram for Carnot
- T constant duering isothermal
- S constant during adiabatic
-> Square
155
AUC in TS diagram for entropy
∫ T dS = ∫ dQ = Q
-> Total heat absorbed
-> = work from 1st law
156
How to prove entropy = 0 in cycle
1. Entropy change = 0 during adiabatic
2. Entropy change for cycle = ∑Isothermal entropy changes
= Cp ∫ dT/T (where T = Tb/Ta) + Cp ∫dT/T (where T = Td - Tc)
3. Collect terms
S = Cp ln (TBTD/TATC)
4. Relates temperatures to pressures using adiabatic T gamma P (1-gamma) = constant for adiabatic
5. Sub temperatures in terms of pressure from 4 into 3
6. PA = PB = PC = PD -> 5. = 0
157
Pressure formulas if problems with piston and spring
1. P1 = Patm + Ppiston
2. P = Patm + Ppiston + Pspring
3. P2 = P1 + ks (V2 - V1)/ A^2
158
Force of spring in piston-cylinder problem
Force linear with volume
Force = ks x = kx (V - V1) / Area
159
Power law (Boltzmann Law)
Power = σ A T^4
σ = Boltzmann constant
160
Boltzmann constant value
5.67x10^-8W/m2K4
161
Emissivity factor
Epsilon - used in Boltzmann law for non black bodies
162
Sum of thermal resistances
∑Ri = 1/A * ∑ Li/κi
κ = thermal conductivity
163
Relationships of temperature through constant section between 2 temperatures
Linear
164
Helmholtz Free Energy formula
F -> F***s ugly skinny”
F = U - TS
165
Best time to use Helmholtz
Constant T and V
166
Helmholtz helps determine…
Maximum work obtainable from a system at constant T
167
Enthalpy formula
H -> “heats you (U) people’s villas”
H = U + PV
168
Enthalpy useful for…
Constant pressure e.g. heat engines
169
Enthalpy explained
Accounts for both internal energy and energy needed to displace system boundaries
170
Gibbs Free Energy formula
G -> Gutless people voted gibbous in
G = U + PV - TS = H - TS
171
Gibbs Free Energy uses
Constant T and P e.g.chemical reactions
172
Gibbs Free Energy useful for
Determining spontaneity of reactaion
- Spontaneous if ΔG<0 (-ve)
173
Grand potential
“Freely open without potential”
Ω = U - TS - μN
μ = chemical potential
174
Grand potential useful for…
Systems in contact with reservoirs that can exchange energy and particles
175
Conjugate variables
- ST
- PV
- N, μ
176
Differential Helmholtz
= dF = - S dT - P dV + μ dN
177
Differential enthalpy
= dH = T dS + V dP + μ N
178
Differential Gibbs free energy
DG = -SdT + VdP + μ dN
179
Potentials dependencies
- F TVs and
- Grab the Ps in
- SUVs
- Have some power
180
“Monotonically” decreasing proof
- Need to show gradient < 0 -> differentiate
181
Relevance of latent heat of fusion ~ heat capacity ratio
- If corresponds to temperature change, heat required for both = same
182
Decrease in Helmholtz in isothermal
Work - upper bound for irreversible
183
Helmholtz in isochoric
Can never increase
184
Helmholtz free energy is a minimum when
System with fixed T and V at equilibrium
185
Gibbs at a minimum
For system with fixed P and T at equilibrium
186
When is enthalpy a measure of heat
Constant pressure
187
What does enthalpy < or> 0 indicate
Endothermic vs exothermic
OPPOSITE - Change in H <0 = exothermic
188
Relevance of free energies being unable to increase?
Sets direction for spontaneity
189
Cv in partial differentials
= (dU/dT)v = T ( dS/dT)v
190
Peak velocity formula
√((2KbT)/M)
191
How to obtain Maxwell- Boltzmann distribution
- Integrate velocity probability distribution F (v) in spherical coordinates (with Jacobian)
192
σ^2 in speed probability
KbT/m
193
Gaussian integral result
Integral from infinity to -infinity of e ^ -(λ x^2 ) = √(π/λ)
194
Gaussian integral used for speed probability MUST KNOW
∫ e ^ - ( x^2/ (2 σ^2)) dx = SQRT ( 2* PI * σ^2
195
Gaussian integral result for 0 to infinity
1/2 * Sqrt ( 2 * PI * σ^2)
196
Relate expectation value of velocity to peak velocity
197
Clausius vs Kelvin Second Law
(Clausius -> “Celsius”, temperature; Kelvin -> “Calvin Klein” -> Hot)
No process is possible whose ONLY result is the
C: transfer of heat from a colder body to a hotter one
K: transformation of heat extracted from a heat source of constant temperature into work
198
Which give out more heat:
- ideal heat pump requiring W work to run
- electric heater converting all work W to heat
Heat pump
199
Joule expansion features
- Adiabatic
- Expands into vacuum
- No heat exchange and no work done
- U remains constant
200
Heat per area from thermal resistance
= Q = ΔT/ thermal resistance
201
Sum of thermal resistance in series
Req = 1/A * Sum of L/K
202
Find entropy of free expansion
- Idealised reversible
- Cancel ‘Vs” in nRT ln (Vf/Vi)
203
Entropy of mixing i.e right chamber with gas A in volume xV, left gas B in (1-x)V
1. P = NRT/V
- n1 = n2 if P and T same for both
2. -> n1/V1 = n2/V2
Let n = n1 + n2 -> solve for n1 and n2
3. Create idealised reversible subbing in values for n1 and n2 in terms of n
204
Entropy of friction
Idealised over temperature change using Cp ln (Tf/Ti)
205
Entropy change of multiple reservoirs
Create intermediate states Tn => T1 = Tstate = dT to Tn = Tres -dT
206
Prove entropy of reservoirs is a minimum
Substitute Tstate/Tres = x and solve entropy equation for different values of x
207
Stirling cycle
Simialr efficiency to Carnot
- 2 isothermal, 2 isochoric
208
Otto cycle
Efficiency based on compression ratio
- 2 adiabatic, 2 isochoric
209
Isochoric heat in Stirling
nCV (Tf-Ti)
210
Work in Stirling
- Work out (expansion) - Work in (compression)
= nRTh ln (Vf-Vi) - nRTc ln (Vf-Vi)
NOT equal as different T values
211
Why similar efficiencies for Stirling and Carnot
- Same idealised reversible manner
- Maximum possible temperature difference governed by temperature difference
212
3 tricks to tackle derivation of speed probability
1. Find derivative of exponential part of Gaussian to simply integral i.e d/dV of e^-(AV^2/B) = -B/A e^-(AV^2/B)
2. Use v*exponent as dv term in integration by parts (NB uv should cancel due to limits)
3. When in 2D or 3D, use v^2 = vx^2 + vy^2 + vz^2 to cancel/change spatial dependence
213
A term in Van Der Waals
Attraction between molecules - reduces effective pressure exerted on wall
214
B term in Van Der Waals
Accounts for size of molecules - reducing total volume
215
When are Van der waals correction terms negligible?
Low densities n/v<<<1
216
Features of Isothermal curves for Van Der Waals (4)
1. Right upper = hottest
2. Dashed line = liquid and vapour equilibrium
3. Thick line = Critical isotherm
4. Dot = critical point
217
Critical isotherm
Isotherm corresponding to critical temperature
218
When are isotherms no longer monotonic?
Below critical temperature -> have local minimum and maximum
219
What can happen at or near critical point in Van Der Waals?
1. Isothermal compressibility value remains negative -> adds to instability (this should be last point)
2. (DP/dV)t > 0 -> instability
If P increases, V increases
3. Negative work provides energy for more pressure fluctuation
220
What does Van Der Waals isotherm being non-monotonic signify?
Phase separation occurring -> liquid to gas
221
How to derive critical temperature and volume of Van Der Waals
1. Differentiate Van Der Waals equation twice ( dP/dV)T
2. Make both = 0 and solve for Vc in terms of b
3. Sub back in to solve for Tc
4. Sub Tc and Vc into Van Der Waals equation to get Pc
222
Prove isothermal compressibility divergence for Van Der Waals
1. Differentiate Van Der Waals equation twice ( dP/dV)T
2. Make both = 0 and solve for Vc in terms of b
3. Sub back in to solve for Tc
4. Sub Tc and Vc into Van Der Waals equation to get Pc
5. Sub all into ideal gas equation
6. (dP/dV)t = 0 so (dV/dP) term in compressibility term diverges
223
Gibbs free energy in VDW below Tc?
Multiply valued
224
What happens to Gibbs Free energy in system at constant T and P?
Minimised
225
What is noticed when plotting pressure on x axis, and Gibbs (G) and V on y?
2 volume values for 1 pressure - reduced to 1 value for G
226
What is relevance of 2 volumes for 1 pressure being represented by 1 G?
Two points can be in equilibrium with each other
227
Metastable state BY
- Supercooled gas
- Lower volume
- Can survive for limited periods though not lowest energy state
228
Metastable state BX
-Superheated liquid
- Can survive for limited periods though not lowest energy state
229
Minimums for Gibbs Free Energy
- Single at high and low pressure
- 2 at critical pressure
230
How is phase coexistence seen from G?
Equal at some points
231
Phase transitions above Tc?
not as distinct
232
Maxwell construction
Finds the pressure at which liquid and gas coexist by ensuring equal G, using AUC on VDW curve
233
Derive phase transition diagram in VDWs
1. Relate G for 2 pressures
G (P1,t ) = G(P0, T) + ∫ (dG/dP)T dP
- limits P1,P0
2. Sub in (dG/dP)t = v
G (P1,t ) = G(P0, T) + ∫ (dG/dP)T dP
- with limits B1,B2
3. G (P1,t ) = G(P0, T) so disappear
∫ V dP = 0
234
Derive internal energy of VDWs
1. From entropy of VDW, find expression by imposing exact differential
2. Sub in dP/dT and P
3. Non zero term = dU/dV -> integrate WRT v
4. Integration constant function of T with constant V
5. Notice n Cv = (dU/dT)v = d/dT (f(T)
And integrate wrt T
6. State answer wiht constant
235
Find entropy of VDW
DS = dQ/T = 1/T (dU + P dV)
U = nCv dT
236
Adiabatic transformations of VDW
1. Find entropy
DS = dQ/T = 1/T (dU + P dV)
U = nCv dT
2. Adiabatic = isentropic
- remove small ‘n’ at start of terms for entropy and make all = consant
*** Ensure divide R by Cv before exponentiating
237
Work by isothermally expanding VDW compared to ideal
- Higher
- Attractive forces reduce pressure and make gas less able to perform work ( less force per unit area)
238
Free expansion of VDW - entropy and temperature
1. Find internal energy
A) From entropy of VDW, find expression by imposing exact differential
B) 2. Sub in dP/dT and P
C) Non zero term = dU/dV -> integrate WRT v
D) Integration constant function of T with constant V
E) Notice n Cv = (dU/dT)v = d/dT (f(T)
And integrate wrt T
2. Equate U final and U initial
3. Rearrange to find Ti/Tf relationship
6. State answer wiht constant
239
Entropy in VDW free compared to ideal
Higher (isothermal + isochoric)
240
Heat loss through longitudinal walls of cylinder - accurate and approximate formulas
Accurate
- Q ∫ dr/r = -2 π L K ∫ dT from separation of variables
Approx
- Q = - 2 π K L (Ri + Rf)/2 * (Tf - Ti)/(Rf -Ri)
241
How to calculate work in piston
Ignore isochoric part - before piston starts moving
- Only isobaric part
243
Sum of thermal resistance in parallel
1/Req = A * Ki/Li
244
Efficient operating between same temperatures as Carnot
Same
245
Method when asked to compare cycle with carnot
Check if can factor out carnot to prove
246
How to construct Jacobian
Crossproduct (determinant if 2d) of partial derivatives
247
Formula for heat pump efficiency
COP heating = Qh/n
COP cooling = Ql/n
248
Diagram for heat pump
- Cooling: Hot reservoir
- Heating: Cold reservoir
- Qh,Qc and W all in same places
249
How to solve for thermal equilibrium Te in heat pump
1. Write efficiency equation in heat,work and temperature
2. Rearrange 1 for Q and equate to other heats
250
Quadratic formula
- b ± √ (b^2 - 4ac) /2a
251
Difference between heater and heat pump if heater 100% efficient
Heater -> all power compensating for heat loss
COP = Qh/W -> Qh = W
252
Formula relating thermal resistance to heat
Q = - ΔT/ Req
253
Formula for power radiation
Pnet = σ e A ΔT^4
σ = boltzmann’s constant, e = emissivity factor (1 for black bodies, <1 for rest, 0 for perfect reflector)
254
2 tricks to use if asked to solve for change or fraction and not all values given
- Simultaneous equations - somethings may cancel
- Check equations, may just have e.g. linear dependence on 1 or 2 factors which can be used to describe relationship/change
255
3 things to remember for packing/huddling penguins
1. For huddle still need side surface are obtained from πr^2 = 1000 * top area
- use this r to solve for sides of cylinder -> approximated as one large cylinder compared to 1000 small ones
2. Power = radiated, but not from bottom
3. The energy saved = 1 - 1/(the number obtained from ratio)
- E.g. If 1000Ap/Ah = 8, Energy saved = 1 - 1/8 = 7/8
256
2 things to remember for packing/huddling penguins
1. For huddle still need side surface are obtained from πr^2 = 1000 * top area
- use this r to solve for sides of cylinder
2. Power = radiated, but not from bottom
257
Quotient rule
VU’ - UV’ / V^2
Where F(x) = U/V
258
Effect on environments of increasing heat transfer with increasing temperature difference
- energy efficiency to ensure morning room temperature is at target
- Hot environments cool faster than cold ones -> need more energy to maintain same temperature
- Most efficient to turn off heat and switch on again later to have right temperature
259
Which Boltzmann constant to use?
- σ = Stefan Boltzmann, thermal radiation only
σ -> S -> shining
Kb -> Kinetic
260
2 ways to increase temperature
- heating
- work
261
Derive differential heat capacity at constant volume
1. Set 1st law in terms of V and T and set dV =0 (in terms of heat - temperature, and volume - argument)
dQ = (dU/dT)v dT + [ (dU/dV)T + P] dV
2. Divide by dT
262
Derive differential heat capacity at constant pressure
1. Set 1st law in terms of P and T and set dP =0
dQ = [ (dU/dT)P + P (dV/dT)P ] dT + [ (dU/dP)T + P (dV/dP)T] dP
2. Divide by dT
263
Where does second term in differential Cp rise from?
Work during heating
264
How to derive differential form of first law
A) Derive differential Cp:
1. Set 1st law in terms of P and T and set dP =0
dQ = [ (dU/dT)P + P (dV/dT)P ] dT + [ (dU/dP)T + P (dV/dP)T] dP
2. Divide by dT
B) Integrate wrt T
- not dQ term!
265
How to derive Cv and Cp relationship
A) Find differential Cp
1. Set 1st law in terms of P and T and set dP =0
dQ = [ (dU/dT)P + P (dV/dT)P ] dT + [ (dU/dP)T + P (dV/dP)T] dP
2. Divide by dT
B) Find differential ideal gas law
1.Ideal gas law and assume 1 mole
2. Take derivatives of both sides wrt variables present
PdV + VdP = RdT
C) Equate PdV in A with PdV in B
D) Set dp = 0
E) Divide by dT
(dQ/dT)P = Cv + R
266
How to derive adiabatic transformations/relations
1. Differential first law (but Q as CvT)
2. Set dQ = 0 as adiabatic
- one side = 0
3. Separation of variables
4. Integrate wrt variables present
5. Log rules to create 1 term
6. Exponentiate
7. Sub in R = Cp - Cv
8. Sub in γ
9. Use ideal gas law to find others
267
Cv for monoatomic
3/2 R
268
Cv for diatomic
5/2 R
269
Cp for monoatomic
5/2 R
270
Cp for diatomic
7/2 R
271
γ formula in Cv and Cp
γ = Cp/Cv
272
Effect on temperature of adiabatic compression
Increases
273
How to find different temperatures etc in cycle?
Adiabatic relations
274
Effect on temperature of adiabatic expansion
Decreases
275
Shape of isothermal curve compared to adiabatic
Isothermal = hyperbola,P proportional to 1/V
Adiabatic steeper decrease, P proportional to 1/V ^ γ
- γ always >1
276
Which molecules have steepest pressure decrease on PV?
monoatomic as γ only 3/2 and pressure proportional to 1/V^ γ
277
Features of heat reservoir
Assumed to be so large it exchanges heat without temperature changes and does no work
278
Alternate name for heat reservoir
Thermostat
279
Location of heat release in Carnot
Isothermal compression
280
Location of heat absorption in Carnot
Isothermal expansion
281
What is Carnot efficiency independent of?
- Type of gas - mono/diatomic, ideal etc
282
Carnot efficiency in words
Efficiency of all reversible engines operating between ONLY 2 reservoirs
Maximal theoretical/ideal efficiency
283
Location of energy exchanges for any and all work in a cycle
Sum of isobaric, isothermal and adiabatic
284
Kelvin-Planck’s LAw
- Alternative second law
No CYCLIC process is possible whose only result is the COMPLETE conversion of HEAT into work
285
Clausius theorem formula
∑Qi/Ti
286
Work for isochoric, isobaric cycle
Top AND bottom - isobaric parts
287
Heat out for isochoric, isobaric cycle
Right AND bottom
288
Heat in for isochoric, isobaric cycle
Top AND left parts
289
How to find adiabatic index from PV given as ratios e.g. P2=5P1
- Reduce to gamma = Ln A/ Ln B
- Find A and B as powers with same base
E.g. if A = 16 and B = 4, A becomes 2^4 and B2^2
- Cancel logs to find ratio
290
Origin of consideration of entropy as new state function
Carnot - Q1/T1 + Q2/T2 = 0
291
Prove entropy = 0 for reversible
1.*unsure if 1. needed - From proof of Clausius theorem
dQ/dt = 0, does not change in cyclic reversible -> suggests state function
2. Draw PV with joining transformations
3. Reverse direction on 1 to create a cycle
4.As cyclic reversible, dQ/T = 0 for 3
5. Separate 4 into 2 reactions and equate
292
Evidence of state function
Unchanged in cyclic reversible transformation
293
Derive adiabatic work formula
1. Sub in P = nRT/V
2. Adiabatic TV relation
- TV^ (γ-1)
- new constant outside integral
3. Integrate
CAREFUL! - not lnV -> add 1 to power
4. Make existing constant = PV^ γ
5. Split V^(1-γ) into V/V^γ so can cancel
294
Find entropy of ideal gas
1. Differential 1st law for 1 mole
2. Sub in P= RT/V and dU = CvdT
3. Solve for dQ
4. dS = dQ/T -> divide 3 by T
5. Separate variables
295
Check entropy is state function
Create exact differential
296
Check heat is not state function
Integrate dS = Cv dT/T + R dV/V wrt T
297
Prove internal energy only temperature dependent
1. 1st law differential as dT and dV
2. Solve for entropy
3. Exact differential
4. (dU/dV)T = 0 i.e. volume independent
298
299
Which 6 factors is efficiency related to?
Qh
QL
TH
TL
Work
r = compression ratio (V2/V1)
300
301
302
What variable do I use “the power an engine produces on the output shaft” as?
Work
303
1 hp in watts
745.7
304
Important to recall about hp
Is converted to watts not joules
305
Qh in engines (words)
Heat absorbed by the engine burning fuel
= heat released or rejected to ambient - production of energy on output shaft
306
307
Rate of fuel consumption formula
M = (Work /efficiency) / energy content of fuel
= QH / energy content of fuel
Energy content = Joules/kg
308
Tricks to solve combined Carnot fridge and engine
1. Equate works
2. Multiply COP and efficiency
3. If asked for ratio, can find something similar and then multiply e.g if looking for Q4/Q1 and have Q3/Q1, multiply by Q4/Q3
309
Steps to prove relation of Carnot efficiency to reservoirs
1. Write efficiency equation
n = ( Qin - lQoutl ) / Qin = ( Qin + Qout ) / Qin = 1 + Qout/Qin
2. Clausius Theorem for reversible
Q1/T1 + Q2/T2 + Q3/T3 = 0
3. Rearrange for Qx - where Qx is the one by itself in 1
4. Add and subtract a factor
- has numerator of 1 and denominator of other with T1/Tfinal included
5. Sub 4 into 1
6. Rearrange as a factor of Carnot efficiency
310
Where to start when expressing something as 2 independent variables e.g. entropy in T and P
Create dV from dT and dP
Same for state e.g. entropy
Sub above into first law
311
How to find state/function in terms of independent variables if not given hint/info
1. Look at 1st law to know what to eliminate - should be differentials of 1 state and possibly 1 variable
-e.g. differential for dV -> not needed as already in equation
2. Find differentials for state need to eliminate and variables of interest
3. Sub into 1st law
DO NOT FORGET THE ‘T’
4. Make exact differentials
- DIFFERENTIATE WITH CONSTANTS AS PER BIG LETTERS, NOT SUBSCRIPTS OR DV/DT/DP -> consider multiplying through by this variable if present if makes differential easier
- If no big letter e.g (dU/dV)T dV to be differentiated wrt T, just change letters in differential
- Check if anything cancels e.g. (dU/dT) = 0
312
Integration variable for isothermal entropy
V
313
Sign of entropy if initial temperature differs from reservoir
Positive
314
If Mass B = 2x Mass A (same substances), what is the ratio of heat capacities?
MB = MA x2
315
How to find entropy from diagonal graph/integration combining 2 variables?
- Need to change variables
- E.g dQ = ( CvdT + PdV)/T = Cv V / R (dT) + P/T dV
1. Sub in dT by differentiating ideal gas law to change to dV
2. Change p in terms of V
3. Need need dP from dT substitution
NB may be able to obtain some variables as y=mx+c e.g. if straight line on PV diagram
316
If AC is a straight line on a PV diagram (A top left, C bottom right) , and you need to find P e.g. as part of entropy calculation, what formula to use?
P = (PAVC - PCVA) / (VC - VA) + (PC - PA)/ (VC - VA)
= AUC / volume
= work / volume
317
How to prove AUC = work in TS diagram?
1. ∫ T.dS = ∫ dQ = Q
2. Q net = (Th - Tc) (S2 - S1)
3. For isothermal:
S2 - S1 = 1/TH ∫dQ = nR ln (Vf/Vi)
4. Work net = nR ( Th - Tc ) ln (V2/V1)
318
Formula for entropy in a cycle
Integral of dQ/T for non adiabatic parts only
319
320
Formula to calculate number of configurations having k outcomes of macro states e.g. k tails on flipping coins
(n) = n!
(k) k! (n - k)!
321
Formula for total possible configurations of macro states
2^n where n = total number of macrostates
322
Probability of obtaining a macro state of X configurations
( n - total number of macrostates) / total number of configurations
( k - X configurations )
E.g. coin with 20 macrostates - possibility of 9 tails:
(20) * 2^-n
(9 )
(n) = n! / [ n! ( n-k)! ]
(k)
323
Prove statistical entropy (Boltzmann law) formula
1. Entropy of free expansion
- > nR ln (1/x)
2. If volume doubles => n R ln 2
3. Kb = R/NA
Change in S = Kb N ln 2
324
Entropy in terms of macrostates (words)
related to probability of a certain macrostates
- related to number of microstates which can give rise to that macrostates
325
Calculate Ω e.g. 100 balls; 50 red, 30 yellow, 20 blue
100!
326
Ω meaning
Number of microstates (configurations) corresponding to a given microstate
327
How to derive entropy from Maxwell relation
1.Differential ideal gas law as dP/dT
- Leave V normal
2. Use Maxwell to add entropy term
3. Temperature a constant as in entropy formula -> separate variables and integrate
328
What is another variable for heat during constant pressure process?
Enthalpy
329
Formula for change in entropy of surroundings from Clausius?
ΔS surr > or = -ΔH/T
330
General relation between Gibbs Free Energy and entropy
When Gibbs increases, total entropy increases
331
Thermal expansivity coefficient in words
Relative increase of volume with temperature
332
Thermal expansivity coefficient formula
α = 1/V (dV/dT) P
333
Isothermal compressibility in words
AKA bulk modulus
Resistance to changes in volume under pressure with constant temperature
334
Isothermal compressibility formula
KT = -1/V (dV/ dP) T
335
How to derive combination of isothermal compressibility and thermal expansivity as partial derivatives (problem sheet 4, Q 3)
1. Differential 1st law
2. Divide by dV (T constant)
3. Use Maxwell relation for F to eliminate entropy
(dS/dV)T = (dP/dT)V
4. Define ratio of expansivity/ compressibility
α/Kt = - (dV/dT)p / (dV/dp)T
4. Reciprocal theorem
- (dV/dT)p / (dV/dp)T = (dV/dT)P (dP/dV)T
5. Reciprocity theorem
(dV/dT)P (dP/dV)T (dP/dV)T = -1
6. α/Kt = (dP/dT)V
-> sub in 3
336
Limits of a Gaussian integral
Plus and minus infinity
337
Limits for expectation value integral
Zero to infinity
338
Value of average velocity from Maxwell distribution in any direction
0
339
What does expectation value mean?
Mean of several velocity measurements
340
Relative velocity sizes - RMS, expectation value or V peak?
Vrms > > V peak
341
Why might I get confused with velocity probability distribution and expectations values?
There is already a v^2 in distribution which I keep forgetting to account for
342
Equipartition theorem
Each translational degree of freedom carries an average energy of KBT/2
343
What indicates degrees of freedom in gases
Top number of Cv/Cp
344
Why do degrees of freedom vary between molecules?
Quantum effects
345
346
How does Cv vary as a function of temperature
-3/2R at low temperatures
- At T_rot jumps to 5/2R
- At T_Vib reaches 7/2R
347
Solid angle formula
Arc area/r^2
348
Explain Maxwell’s Demon
Paradox:
- By moving fast particles to one side and slow to the other with trapdoor, could create temperature difference without work
Resolution:
- Demon’s processing to sort particles require energy and increases entropy
349
Effusion rate in words
Rate of particles flowing out of a region per unit area
350
Derive effusion rate
1. Ring of θ + d θ in sphere with hole where particles leave projected through centre
2. Area of ring = 2π sinθ dθ
- θ is between horizontal (which passing though hole and centre of ring and sphere, and hypotenuse to edge of ring)
- Perpendicular line from horizontal to ring = sinθ
3. Molecules moving in any direction = 4π
4. Fraction of molecules with direction between θ and dθ = 2/3
= 2π sinθ dθ/4π
=sinθ dθ/2
5. Number density (n/V) * 4
Γ(v,θ) = N/V v cosθ f(v) dv 1/2 sinθ dθ
6. Integrate 5 in spherical to obtain effusion rate
351
Value for effusion from integration at end of derivation
1/4 * n (number density)
352
Maxwell-Boltzmann normalisation constant, C
1/2 [ (m / (KbT) ]^2
353
Difference between value for velocity distribution of gas escaping through effusion hole and Maxwell-Bolztmann and physical meaning
- Additional power of v (velocity) as particles obey effusion rate formula
- Particles which can escape are going faster than normal
354
How to calculate expected kinetic energy of gas escaping from hole in effusion
- Use f_esc formula and 1/*m* ^2 and integrate
(Should start with 5 ‘v’s)
355
Value for E_esc -> expected kinetic energy of effusion gas
356
Ratio of kinetic energy to
E_esc = 4/3
357
Assumption about collisions
All elastic - momentum and energy conserved
358
Formula for momentum change on boundary due to collision of particle at angle θ
Δ P = 2mv cos θ
359
When is angle of collision 0
Head on
360
Range of particle velocity for collisions
-V to +V
361
Formula for gas pressure derived from momentum
P = dv dθ momentum exchange per collision x collision rate
= dv dθ [ ΔP x Γ(v,θ) ]
= N/V ∫ dv ∫ dθ ( 2mv cosθ) x vcosθ f(v) 1/2 sinθ
- Integrate from 0 to infinity and 0 to π/2
362
Pressure value when derived from momentum
P = 1/3 N/V m
363
How to obtain ideal gas law from pressure value when derived from momentum
Replace with 3KbT/M
364
Meaning of no mass in ideal gas equation
Gas pressure does not depend on mass - only number density and temperature
365
Paradox of gas pressure independent of mass
- Larger particles should have more momentum -> more pressure
- Larger particles need more energy to reach given temperature than smaller ones
366
How to verify Graham’s Law
1. Expectation value ->
= √[(8KbT)/(πm)
2. Insert into equation for effusion (Φ =1/2 n) and sub in n=N/V
3. Collect leaving N/V alone outside squareroot
4. Sub in PV = nRT
Φ = P / √(2 πMKbT)
367
Reciprocity theory formula from notes
(dA/dB)C * (dB/dC)A = - (dA/dC)B
368
Formula for VDW entropy comes from…
Isochoric heating and isothermal expansion
1: nCv ln (Tf/Ti) + nR ln (Vf -Nb/ VI -nb)
369
370
Evaluating temperature dependence at phase boundaries
1. No volume change -> d/dT as partial differenctials (no V!)
2. Entropy vaoupr - entropy liquido
371
Latent heat formula in terms of entropy
L = ΔQrev = Tc (S2 - S1)
372
Formula for gradient of entropy with temperature as related to heat capacity
C_x = T (dS/dT)_x
373
What happen to entropy curve at phase transition?
Jumps - long vertical line
374
Trouton’s rule in words
Describes relationship between latent heat of vaporisation and boiling point temperature (approximately proportional)
375
Clausius-Clapeyron Equation in words
Describes slope of phase boundary in PT diagram as purely determined by:
- latent heat
- temperature at phase boundary
- difference in volume between 2 phases
376
Clausius-Clapeyron formula
Describes slope of phase boundary in PT diagram as purely determined by:
- latent heat
- temperature at phase boundary
- difference in volume between 2 phases
377
Trouton’s Law formula
L_vapour = AT_b
A = Constant, usually 8-10kJ/mol.K
T_b = boiling temperature
378
Why is latent heat slightly higher than expected?
L involves contribution for attractive intermolecular potential
379
Which 2 molecules are outliers for Trouton’s law and why?
- He -> quantum
- H20 -> polar with dipole moment so different intermolecular potential
380
Chemical potential in words
Gibbs free energy per particle
381
Prove chemical potential, μ, is in both phases at equilibrium
1. Chemical potential of Gibbs Free
dG = VdP - SdT + μdN
2. N1 particles are in equilibrium with N2:
G_tot = N1 μ1 + N2 μ2
3. At equilibrium dG = 0
-> dN1 = -dN2
=> μ1 = μ2
382
Derive Clausius-Clapeyron equation
1. Describe phase boundary in PT plane
μ1 (p,T) = μ2 (p, T)
2. As move along phase boundary:
μ1 (p+dp, T + dT) = μ2 (p + dp, T + dT)
=> d μ1 = d μ2
3. Differentiate μ
- d μ = -SdT + VdP
=> -S1dT + v1dP = -S2dT + v2dP
4. Rearrange
(dP/dT) = (S2 - S1)/ (v2 - v1)
5. Define latent heat per particle, l as TΔS
(dP/dT) = l (small l) / T(v2 - v1)
6. Total:
(dP/dT) = L / T(V2 - V1)
383
Derive phase boundary equation
1. Clausius-Clapeyron
- dp/dt = L / [T (V2-V1)]
2. Assume V_vapour>>VLiquid so single entity
- dp/dt = L / (TV)
3. Sub in ideal gas law -> 1/V = P/nRT
- dp/dt = LP / (RT^2)
4. Rearrange for dP/P (separate variables)
dP/P = LdT/ ( RT^2)
5. Integrate
-> ln P = -L/RT + constant
6. Phase boundary equation:
p = p_o e^ (-L/RT)
384
What does ΔV on melting being relatively small at phase boundary result in?
PT gradient very steep - small change in T -> high change in p
385
Unusual phase properties of water due to polarity
- Ice has lower density (floats)
- Shrinks on melting leading to negative gradient of liquid-solid coexistence
- When pressure increases, ice melts
386
Mole fraction x
Number of moles/total number of moles
387
Gibbs phase rule in words
How many independent variables that we can freely control in system that is in equilibrium
388
Gibbs phase rule formula
F = [ P ( C - 1 ) + 2 ] - C (P - 1) -> simplifies to:
F = C - P + 2
F = number of degrees of freedom
P = Number of phases - can include e.g. paramagnetic, superconducting, ferromagnetic etc
C = Number of components - each component can be in any one of the phases
389
Gibbs phase rule for single component system
C = 1, F = 3-P
- 1 phase -> F =2 and whole plane accessible
- 2 phases -> F = 1 and 2 phases can only coexist on common line
- 3 phases -> F = 0, can only coexist at common point
390
Gibbs phase rule for two component system
C = 2, F = 4 - P
- Fix pressure so remaining degrees of freedom F’ = F -1 = 3 - P
- Rest as for 1 component
391
Colligative properties
Depends only on number of solute particles, not type or identity
392
Must remember when deriving for VDW gases
DO NOT USE IDEAL GAS EQUATION
393
Formula for reservoir change entropy CORRECT
MC/Tres * (Ti - Tres)
Where Ti = temp of previous body in chain, even if other reservoir e.g. water absorbed
394
How is entropy of sample of water affected by addition of intermediate reservoirs
unchanged
395
How is total entropy of reservoirs affected by adding more intermediate reservoirs
Entropy of reservoirs more negative with more reservoirs
396
Finding optimal intermediate reservoir temperature to minimise entropy
d/dT of reservoir entropies only (no LN term from water/gas)
397
Integration value of v f(v) dv
398
Real m^3 to litres
Same
